Optimal. Leaf size=36 \[ \frac{\text{csch}(a-c) \log (\cosh (a+b x))}{b}-\frac{\text{csch}(a-c) \log (\cosh (b x+c))}{b} \]
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Rubi [A] time = 0.0246153, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5644, 3475} \[ \frac{\text{csch}(a-c) \log (\cosh (a+b x))}{b}-\frac{\text{csch}(a-c) \log (\cosh (b x+c))}{b} \]
Antiderivative was successfully verified.
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Rule 5644
Rule 3475
Rubi steps
\begin{align*} \int \text{sech}(a+b x) \text{sech}(c+b x) \, dx &=\text{csch}(a-c) \int \tanh (a+b x) \, dx-\text{csch}(a-c) \int \tanh (c+b x) \, dx\\ &=\frac{\text{csch}(a-c) \log (\cosh (a+b x))}{b}-\frac{\text{csch}(a-c) \log (\cosh (c+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.206235, size = 27, normalized size = 0.75 \[ \frac{\text{csch}(a-c) (\log (\cosh (a+b x))-\log (\cosh (b x+c)))}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 77, normalized size = 2.1 \begin{align*} -2\,{\frac{\ln \left ({{\rm e}^{2\,bx+2\,a}}+{{\rm e}^{2\,a-2\,c}} \right ){{\rm e}^{a+c}}}{b \left ({{\rm e}^{2\,a}}-{{\rm e}^{2\,c}} \right ) }}+2\,{\frac{\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ){{\rm e}^{a+c}}}{b \left ({{\rm e}^{2\,a}}-{{\rm e}^{2\,c}} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6957, size = 92, normalized size = 2.56 \begin{align*} \frac{2 \, e^{\left (a + c\right )} \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} - \frac{2 \, e^{\left (a + c\right )} \log \left (e^{\left (-2 \, b x\right )} + e^{\left (2 \, c\right )}\right )}{b{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87307, size = 490, normalized size = 13.61 \begin{align*} \frac{2 \,{\left ({\left (\cosh \left (-a + c\right ) - \sinh \left (-a + c\right )\right )} \log \left (\frac{2 \,{\left (\cosh \left (b x + c\right ) \cosh \left (-a + c\right ) - \sinh \left (b x + c\right ) \sinh \left (-a + c\right )\right )}}{\cosh \left (b x + c\right ) \cosh \left (-a + c\right ) -{\left (\cosh \left (-a + c\right ) + \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right ) + \cosh \left (b x + c\right ) \sinh \left (-a + c\right )}\right ) -{\left (\cosh \left (-a + c\right ) - \sinh \left (-a + c\right )\right )} \log \left (\frac{2 \, \cosh \left (b x + c\right )}{\cosh \left (b x + c\right ) - \sinh \left (b x + c\right )}\right )\right )}}{b \cosh \left (-a + c\right )^{2} - 2 \, b \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + b \sinh \left (-a + c\right )^{2} - b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}{\left (a + b x \right )} \operatorname{sech}{\left (b x + c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22999, size = 135, normalized size = 3.75 \begin{align*} \frac{2 \, e^{\left (a + c\right )} \log \left (\frac{{\left | -{\left | e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )} \right |} + 2 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )} \right |}}{{\left | e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )} \right |} + 2 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}}\right )}{b{\left | e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )} \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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